Multiplier circuit - analytical expression

[nicrip]

I have been assigned a task that I have found to be very difficult - to try and develop an alaytical (mathematical) expression for the output voltage with respect to time of the cockroft-walton voltage doubler shown below (so that I can plot the transient response), when the input voltage is a sinusoidal source. I was wondering if anybody would know how this could be done, or if such a expression has already been developed. The problem is that the diodes continually switch and charge the caps, but I cannot assume that the caps (C1) would charge to the peak input voltage in one cycle.

Similarly with the rectifier circuit below (assume the resistor is replaced by a capacitor), how would I develop a mathematical expression for the output voltage if you cannot assume that the cap voltage reaches the peak input voltage in one cycle? Would I somehow have to find the points at which the diode switches, and continually calculate the voltage on the cap as it gains voltage over several cycles of the input voltage? I appreciate any help.

 

[Proton Soup]

it's not as if there is a pretty equation. the devices are non-linear, and so simulators like SPICE solve them numerically, which takes considerably more time to solve than linear circuits.